nLab
maximal sieve

A sieve over an object cc of category CC is the maximal sieve over cc if it contains all morphisms with targetcc. In other words, it agrees with the class of all objects of the slice categoryC/cC/c. One of the axioms of Grothendieck topologies says that any maximal sieve (that is the maximal sieve for any object in CC) is a covering sieve.

A maximal sieve is any sieve generated by an identity morphism in CC (recall that a sieve generated by a family of morphisms {gi:ci→c}i∈I\{g_i:c_i\to c\}_{i\in I} is the class of all morphisms of the form gi∘hg_i\circ h where h:e→cih:e\to c_i is a morphism in CC). If a sieve over cc is considered as a subpresheaf of the representable presheaf Hom(−,c)Hom(-,c), then a sieve is maximal iff it is Hom(−c)Hom(-c).